an:06878476
Zbl 1388.57010
Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, Vivek Kumar
Colored HOMFLY polynomials of knots presented as double fat diagrams
EN
J. High Energy Phys. 2015, No. 7, Paper No. 109, 70 p. (2015).
00402133
2015
j
57M27 57R56 81T45
quantum groups; Chern-Simons theories; topological field theories
Summary: Many knots and links in \(S^{3} \) can be drawn as gluing of three manifolds with one or more four-punctured \(S^2\) boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of \textit{four}-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of \textit{J. Gu} and \textit{H. Jockers} [Commun. Math. Phys. 338, No. 1, 393--456 (2015; Zbl 1328.81193)], the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with \(A\)-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.
Zbl 1328.81193